A Random Differential Equation Approach to the Probability Distribution of Bod and Do in Streams

In this paper a stochastic model for stream pollution is given which involves a random differential equation of the form \[( * )\qquad \dot {\bf X}( t ) = {\bf A}{\bf X}( t ) + {\bf Y},\quad t\geqq 0,\] where ${\bf X}( t )$ is a two-dimensional vector-valued stochastic process with the first component giving the biochemical oxygen demand (BOD) and the second component representing the dissolved oxygen (DO) at distance t downstream from the source of pollution. The fundamental Liouville’s theorem is utilized to obtain the probability distribution of the solution of $( * ),{\bf X}( t )$, at each t with various distributional assumptions on the random initial conditions and random inhomogeneous term. Computer simulations of the trajectories of the BOD and DO processes as well as the mean and variance functions are given for several initial distributions and are compared with the deterministic results.